DOCSLIB.ORG

Chapter I Set Theory

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Chapter I Set Theory Chapter I Set Theory There is surely a piece of divinity in us, something that was before the elements, and owes no homage unto the sun. Sir Thomas Browne One of the benefits of mathematics comes from its ability to express a lot of information in very few symbols. Take a moment to consider the expression d sin( θ). dθ It encapsulates a large amount of information. The notation sin( θ) represents, for a right triangle with angle θ, the ratio of the opposite side to the hypotenuse. The differential operator d/dθ represents a limit, corresponding to a tangent line, and so forth. Similarly, sets are a convenient way to express a large amount of information. They give us a language we will find convenient in which to do mathematics. This is no accident, as much of modern mathematics can be expressed in terms of sets. 1 2 CHAPTER I. SET THEORY 1 Sets, subsets, and set operations 1.A What is a set? A set is simply a collection of objects. The objects in the set are called the elements . We often write down a set by listing its elements. For instance, the set S = 1, 2, 3 has three elements. Those elements are 1, 2, and 3. There is a special symbol,{ }, that we use to express the idea that an element belongs to a set. For instance, we∈ write 1 S to mean that “1 is an element of S.” For∈ the set S = 1, 2, 3 , we have 1 S, 2 S, and 3 S. We can write this more quickly as: 1 , 2{, 3 S}. We can express∈ the∈ fact that 4∈ is not an element of S by writing 4 / S. ∈ ∈ Example 1.1. Let P be the set 16 , 5, 2, 6, 9 . Is 6 P ? Yes! Is 5 P ? No, so we write 5 / P . { − } ∈ ∈ ∈ △ The order of the elements in a set does not matter, so we could have written S = 1, 2, 3 as S = 1, 3, 2 , or as S = 3, 2, 1 . If an element is repeated in a set, we do{ not count} the{ multiplicity.} Thus {1, 2, 3}, 1 is the same set as S = 1, 2, 3 . We say that two sets are equal when they{ have exactly} the same elements. { } Not all sets consist of numbers. For instance T = a, b, c, d is a set whose elements are the letters a, b, c, d . Sets may have words, names,{ symbols,} and even other sets as elements. Example 1.2. Suppose we want to form the set of Jesus’ original twelve apostles. This would be the set Apostles = Peter , James , John the beloved ,..., Judas Iscariot . { } We put the 3 dots in the middle to express the fact that there are more elements which we have not listed (perhaps to save time and space). △ The Last Supper , ca. 1520, by Giovanni Pietro Rizzoli. The next example is a set with another set as an element. 1. SETS, SUBSETS, AND SET OPERATIONS 3 Example 1.3. Let S = 1, 5, 4, 6 , 3 . This set has four elements. We have 1, 5, 3, 4, 6 S, but 4 / S{. However,{ } 4 } 4, 6 and 4, 6 S. { } ∈ ∈ ∈ { } { } ∈ △ It can be confusing when sets are elements of other sets. You might ask why mathematicians would allow such confusion! It turns out that this is a very useful thing to allow; just like when moving, the moving truck (a big box) has boxes inside of it, each containing other things. Advice 1.4. You can think of sets as boxes with objects inside. So we could view the set 1, 5, 4, 6 , 3 from the previous example as the following box, which contains{ another{ } box:} 1 5 4 6 3 Figure 1.4 : A box with a box inside, each containing some numbers. 1.B Naming sets We’ve seen that capitalized Roman letters can be used to give names to sets. Some sets are used so often that they are represented by special symbols. Here are a couple of examples. The set of natural numbers is the set • N = 1, 2, 3,... { } This is the first example we’ve given of an infinite set , i.e., a set with infinitely many elements. The set of integers is • Z = ..., 3, 2, 1, 0, 1, 2, 3, 4,... { − − − } The dots represent the fact that we are leaving elements unwritten in both directions. We use the fancy letter “ Z” because the word “integer” in German is “Zahlen.” Some sets are constructed using rules. For example, the set of even integers can be written as ..., 4, 2, 0, 2, 4,... { − − } but could also be written in the following ways: (1.5) 2x : x Z { ∈ } (1.6) x Z : x is an even integer { ∈ } (1.7) x : x = 2 y for some y Z . { ∈ } 4 CHAPTER I. SET THEORY We read the colon as “such that,” so ( 1.5 ) is read as “the set of elements of the form 2x such that x is an integer.” Writing sets with a colon is called set-builder notation . Notice that x Z : 2 x + 1 { ∈ } doesn’t make any sense, since “2 x + 1” is not a condition on x. Here are a few more examples. The set of prime numbers is 2, 3, 5, 7, 11 , 13 ,... = x N : x is prime . { } { ∈ } Similarly, Apostles = x : x was one of the original 12 apostles of Jesus . { } With set-builder notation, we can list a few more very important sets. The set of rational numbers is • Q = a/b : a, b Z, b = 0 . { ∈ 6 } Note that there is no problem with the fact that different fractions can represent the same rational number, such as 1 /2 = 2 /4. Repetitions do not matter in sets. We will occasionally need the fact that we can always write a rational number as a fraction a/b in lowest terms : i.e., so that a and b have no common factor larger than 1. We will prove this in Section 18 (see Exercise 18.3 ). The set of real numbers is • R = x : x has a decimal expansion . { } So we have π = 3 .14159 . R, 3 = 3 .00000 . R, and √2 R. Later in ∈ ∈ ∈ this book we will prove √2 / Q. The set of complex numbers ∈is • C = a + bi : a, b R, i 2 = 1 . { ∈ − } Is 3 a complex number? Yes, because we can take a = 3 and b = 0. So we have 3 N, 3 Z, 3 Q, 3 R, and 3 C! ∈ ∈ ∈ ∈ ∈ Example 1.8. Which of the named sets does π = 3 .14159 . belong to? We have π R and π C. On the other hand, since 3 < π < 4 we have π / N and π / Z. It ∈ ∈ ∈ ∈ is true, but much harder to show, that π / Q. ∈ △ There is one more set we will give a special name. The empty set is the set with no elements. We write it as = . • ∅ { } Warning 1.9. The empty set is not nothing . It has no elements, but the empty set is something . Namely, it is “the set with nothing in it”. Thinking in terms of boxes, we can think of the empty set as an empty box. The box is something even if it has nothing in it. The symbol does not mean nothing. It means . ∅ { } 1. SETS, SUBSETS, AND SET OPERATIONS 5 Example 1.10. Sometimes we want the empty set to be an element of a set. For instance, we might take S = . {∅} The set S has a single element, namely . We could also write S = . In terms of boxes, S is the box containing an empty∅ box. Note that not all sets{{ have }} the empty set as an element. △ A box with an empty box inside, representing . {∅} 1.C Subsets In many activities in life we don’t focus on all the elements of a set, but rather on subcollections. To give just a few examples: The set of all phone numbers is too large for most of us to handle. The subcol- • lection of phone numbers of our personal contacts is much more manageable. If we formed the set of all books ever published in the world, this set would be • very large (but still finite!). However, the subcollection of books we have read is much smaller. If we want to count how many socks we own, we could use elements of the • integers Z, but since we cannot own a negative number of socks, a more natural set to use would be the subcollection of nonnegative integers Z = 0, 1, 2, 3,... ≥0 { } A subcollection of a set is called a subset . When A is a subset of B we write A B and if it is not a subset we write A * B. There are a couple different ways to think⊆ about the concept of A being a subset of B. Option 1: To check that A is a subset of B, we check that every element of A also belongs to B. Example 1.11. (1) Let A = 1, 5, 6 and B = 1, 5, 6, 7, 8 . Is A a subset of B? Yes, because we can check{ that each} of A’s three{ elements,} 1, 5, and 6, belongs to B. (2) Let A = 6, 7/3, 9, π and B = 1, 2, 6, 7/3, π, 10 . Is A B? No, because { } { } ⊆ 9 A but 9 / B. So we write A * B. ∈ ∈ (3) Let A = N and B = Z. Is A B? Yes, every natural number is an integer.
Load more

Recommended publications
  • Sets, Functions
    Sets 1 Sets Informally: A set is a collection of (mathematical) objects, with the collection treated as a single mathematical object. Examples: • real numbers, • complex numbers, C • integers, • All students in our class Defining Sets Sets can be defined directly: e.g. {1,2,4,8,16,32,…}, {CSC1130,CSC2110,…} Order, number of occurence are not important. e.g. {A,B,C} = {C,B,A} = {A,A,B,C,B} A set can be an element of another set. {1,{2},{3,{4}}} Defining Sets by Predicates The set of elements, x, in A such that P(x) is true. {}x APx| ( ) The set of prime numbers: Commonly Used Sets • N = {0, 1, 2, 3, …}, the set of natural numbers • Z = {…, -2, -1, 0, 1, 2, …}, the set of integers • Z+ = {1, 2, 3, …}, the set of positive integers • Q = {p/q | p Z, q Z, and q ≠ 0}, the set of rational numbers • R, the set of real numbers Special Sets • Empty Set (null set): a set that has no elements, denoted by ф or {}. • Example: The set of all positive integers that are greater than their squares is an empty set. • Singleton set: a set with one element • Compare: ф and {ф} – Ф: an empty set. Think of this as an empty folder – {ф}: a set with one element. The element is an empty set. Think of this as an folder with an empty folder in it. Venn Diagrams • Represent sets graphically • The universal set U, which contains all the objects under consideration, is represented by a rectangle.
    [Show full text]
  • The Universal Finite Set 3
    THE UNIVERSAL FINITE SET JOEL DAVID HAMKINS AND W. HUGH WOODIN Abstract. We define a certain finite set in set theory { x | ϕ(x) } and prove that it exhibits a universal extension property: it can be any desired particular finite set in the right set-theoretic universe and it can become successively any desired larger finite set in top-extensions of that universe. Specifically, ZFC proves the set is finite; the definition ϕ has complexity Σ2, so that any affirmative instance of it ϕ(x) is verified in any sufficiently large rank-initial segment of the universe Vθ ; the set is empty in any transitive model and others; and if ϕ defines the set y in some countable model M of ZFC and y ⊆ z for some finite set z in M, then there is a top-extension of M to a model N in which ϕ defines the new set z. Thus, the set shows that no model of set theory can realize a maximal Σ2 theory with its natural number parameters, although this is possible without parameters. Using the universal finite set, we prove that the validities of top-extensional set-theoretic potentialism, the modal principles valid in the Kripke model of all countable models of set theory, each accessing its top-extensions, are precisely the assertions of S4. Furthermore, if ZFC is consistent, then there are models of ZFC realizing the top-extensional maximality principle. 1. Introduction The second author [Woo11] established the universal algorithm phenomenon, showing that there is a Turing machine program with a certain universal top- extension property in models of arithmetic.
    [Show full text]
  • Sets and Functions
    Unit SF Sets and Functions Section 1: Sets The basic concepts of sets and functions are topics covered in high school math courses and are thus familiar to most university students. We take the intuitive point of view that sets are unordered collections of objects. We first recall some standard terminology and notation associated with sets. When we speak about sets, we usually have a “universal set” U in mind, to which the various sets of our discourse belong. Definition 1 (Set notation) A set is an unordered collection of distinct objects. We use the notation x ∈ S to mean “x is an element of S” and x∈ / S to mean “x is not an element of S.” Given two subsets (subcollections) of U, X and Y , we say “X is a subset of Y ,” written X ⊆ Y , if x ∈ X implies that x ∈ Y . Alternatively, we may say that “Y is a superset of X.” X ⊆ Y and Y ⊇ X mean the same thing. We say that two subsets X and Y of U are equal if X ⊆ Y and Y ⊆ X. We use braces to designate sets when we wish to specify or describe them in terms of their elements: A = {a, b, c}, B = {2, 4, 6,...}. A set with k elements is called a k-set or set with cardinality k. The cardinality of a set A is denoted by |A|. Since a set is an unordered collection of distinct objects, the following all describe the same 3-element set {a, b, c} = {b, a, c} = {c, b, a} = {a, b, b, c, b}.
    [Show full text]
  • Church's Set Theory with a Universal
    Church’s Set Theory with a Universal Set Thomas Forster Contents 1 Rieger-Bernays permutation models 2 2 Church-Oswald models 6 2.1 Applications of the technique . 9 2.1.1 Models of NF2 ........................ 9 2.1.2 An elementary example . 11 2.1.3 P-extending models of Zermelo to models of NFO . 14 2.1.4 Church’s Model . 17 2.2 Wellfounded sets in CO-structures . 20 3 Open problems 26 3.1 The Axioms of Sumset and Power set . 26 3.2 Natural strengthenings of theorem 22 . 26 3.3 Axiomatisability . 27 3.4 Constructive CO constructions . 27 3.5 Schr¨oder-Bernstein . 28 3.6 Mitchell’s set theory . 29 3.7 Extensional quotients? . 29 Abstract A detailed and fairly elementary introduction is given to the techniques used by Church to prove the consistency of his set theory with a universal set by constructing models of it from models of ZF. The construction is explained and some general facts about it proved. Introduction In 1974 Alonzo Church and Urs Oswald simultaneously and independently lit upon a refinement of Rieger-Bernays permutation models which enabled them to give elementary proofs of consistency for some set theories with a universal set. My own interest began much later, when I made the aquaintance of Flash Sheridan, a former student of Church’s who was then writing an Oxford D.Phil. 1 thesis on Church’s work in this area. I am greatly endebted to Flash for kindling my interest in this engaging bywater, and for showing me some relevant and (even now) unpublished material.
    [Show full text]
  • Consistent Set Theories with Universal Set
    IV CONSISTENT SET THEORIES WITH UNIVERSAL SET On occasion of the 1971 Berkeley symposium celebrating Alfred Tarski’s achievements in logic and algebra Alonzo Church, who otherwise did not work much in set theory, presented a new system of set theory (Church 1974). Church saw the two basic assumptions of post-naïve set theories in a restriction of comprehension to a form of separation (as in ZFC) and in a limitation of size (as in NBG). Similar to the criticism levelled against Limitation of Size in chapter II above Church regarded Limitation of Size as ad hoc (against the antinomies) and ‘never well supported’ as it proclaims a stopping point of further structures although classes are introduced (in NBG and MK) as collections, which can be quantified over. Church’s set theory – let us call it “CST” here – follows ZFC in its idea of separation, but allows for collections that are ‘large’ in a way that even the larger transfinite sets of ZFC are not. CST does not introduce classes, but introduces a distinction within the area of sets. It allows even for U = {x | x=x}. CST distinguishes between ‘low sets’, which have a 1:1-relation to a well- founded set, ‘high sets’, which are (absolute) complements of low sets and ‘intermediate’ sets which are neither. These labels pertain to the cardinality of sets. High sets are in 1:1-correspondence to the universal set U, low sets never. Because of the CST version of the Axiom of Choice for a set x which is not low every ordinal has a 1:1-relation to some subset of x.
    [Show full text]
  • Section 6.1 Sets and Set Operations
    Section 6.1 Sets and Set Operations Sets A set is a well-defined collection of objects. The objects in this collection are called elements of the set. If a is an element of the set A then we write a A,ifa is not an element of a set A,thenwe 2 write a/A. 2 Roster and Set-Builder Notation Roster notation will be used most commonly in this class, and consists of listing the elements of a set in between curly braces. Set-builder notation is when a rule is used to define a definite property that an object must have in order to be in the set. 1. Let A be the set of all letters in the English alphabet. (a) Write A in roster notation and in set-builder notation. (b) Is the greek letter β an element of A? Set Equality Two sets A and B are equal,writtenA = B,ifandonlyiftheyhaveexactlythe same elements. 2. Let A = a, e, l, t, r . Which of the following sets are equal to A?(Chooseallthatapply.) { } (a) x xisaletterofthewordlatter { | } (b) x xisaletterofthewordlater { | } (c) x xisaletterofthewordlate { | } (d) x xisaletterofthewordrated { | } (e) x xisaletterofthewordrelate { | } Subset If every element of a set A is also an element of a set B,thenwesaythatA is a subset of B and we write A B. ✓ Note: If we write A B,thenthismeansthatA is a proper subset of B,withoutthepossibility ⇢ of equality. Therefore, for any set A, A is NOT a proper subset of itself. 3. If A = u, v, y, z and B = x, y, z ,determinewhetherthefollowingstatementsaretrueorfalse.
    [Show full text]
  • An Introduction to Elementary Set Theory
    An Introduction to Elementary Set Theory Guram Bezhanishvili and Eachan Landreth∗ 1 Introduction In this project we will learn elementary set theory from the original historical sources by two key figures in the development of set theory, Georg Cantor (1845{1918) and Richard Dedekind (1831{ 1916). We will learn the basic properties of sets, how to define the size of a set, and how to compare different sizes of sets. This will enable us to give precise definitions of finite and infinite sets. We will conclude the project by exploring a rather unusual world of infinite sets. Georg Cantor, the founder of set theory, considered by many as one of the most original minds in the history of mathematics, was born in St. Petersburg, Russia in 1845. His parents moved the family to Frankfurt, Germany in 1856. Cantor entered the Wiesbaden Gymnasium at the age of 15, and two years later began his university career in Z¨urich, Switzerland. In 1863 he moved to the University of Berlin, which during Cantor's time was considered the world's leading center of mathematical research. Four years later Cantor received his doctorate under the supervision of the great Karl Weierstrass (1815{1897). In 1869 Cantor obtained an unpaid lecturing post at the University of Halle. Ten years later he was promoted to a full professor. However, Cantor never achieved his dream of holding a Chair of Mathematics at Berlin. It is believed that one of the main reasons was the nonacceptance of his theories of infinite sets by the leading mathematicians of that time, most noticeably by Leopold Kronecker (1823{1891), a professor at the University of Berlin and a very influential figure in German mathematics, both mathematically and politically.
    [Show full text]
  • The Axiom of Determinacy
    Virginia Commonwealth University VCU Scholars Compass Theses and Dissertations Graduate School 2010 The Axiom of Determinacy Samantha Stanton Virginia Commonwealth University Follow this and additional works at: https://scholarscompass.vcu.edu/etd Part of the Physical Sciences and Mathematics Commons © The Author Downloaded from https://scholarscompass.vcu.edu/etd/2189 This Thesis is brought to you for free and open access by the Graduate School at VCU Scholars Compass. It has been accepted for inclusion in Theses and Dissertations by an authorized administrator of VCU Scholars Compass. For more information, please contact [email protected]. College of Humanities and Sciences Virginia Commonwealth University This is to certify that the thesis prepared by Samantha Stanton titled “The Axiom of Determinacy” has been approved by his or her committee as satisfactory completion of the thesis requirement for the degree of Master of Science. Dr. Andrew Lewis, College of Humanities and Sciences Dr. Lon Mitchell, College of Humanities and Sciences Dr. Robert Gowdy, College of Humanities and Sciences Dr. John Berglund, Graduate Chair, Mathematics and Applied Mathematics Dr. Robert Holsworth, Dean, College of Humanities and Sciences Dr. F. Douglas Boudinot, Graduate Dean Date © Samantha Stanton 2010 All Rights Reserved The Axiom of Determinacy A thesis submitted in partial fulfillment of the requirements for the degree of Master of Science at Virginia Commonwealth University. by Samantha Stanton Master of Science Director: Dr. Andrew Lewis, Associate Professor, Department Chair Department of Mathematics and Applied Mathematics Virginia Commonwealth University Richmond, Virginia May 2010 ii Acknowledgment I am most appreciative of Dr. Andrew Lewis. I would like to thank him for his support, patience, and understanding through this entire process.
    [Show full text]
  • Russell, His Paradoxes, and Cantor's Theorem: Part I
    Philosophy Compass 5/1 (2010): 16–28. Russell, His Paradoxes, and Cantor’s Theorem: Part I Kevin C. Klement∗ University of Massachusetts Abstract In these articles, I describe Cantor’s power-class theorem, as well as a number of logical and philosophical paradoxes that stem from it, many of which were discovered or considered (implic- itly or explicitly) in Bertrand Russell’s work. These include Russell’s paradox of the class of all classes not members of themselves, as well as others involving properties, propositions, descriptive senses, class-intensions, and equivalence classes of coextensional properties. Part I focuses on Can- tor’s theorem, its proof, how it can be used to manufacture paradoxes, Frege’s diagnosis of the core difficulty, and several broad categories of strategies for offering solutions to these paradoxes. 1. Introduction In 1961, W. V. Quine described a philosophical method he dubbed ‘The Ways of Para- dox’. It begins with a seemingly well-reasoned argument leading to an apparently ab- surd conclusion. It continues with careful scrutiny of the reasoning involved. If we are ultimately unwilling to accept the conclusion as justified, the process may end with the conclusion that, ‘some tacit and trusted pattern of reasoning must be made explicit and be henceforward avoided or revised’ (‘Ways’ 11). This method perfectly describes Bertrand Russell’s philosophical work, especially from 1901 to 1910, while composing Principia Mathematica. Russell is most closely associated with the class-theoretic antinomy bearing his name: the class of all those classes that are not members of themselves would appear to be a member of itself if and only if it is not.
    [Show full text]
  • Sets and Venn Diagrams
    The Improving Mathematics Education in Schools (TIMES) Project NUMBER AND ALGEBRA Module 1 SETS AND VENN DIAGRAMS A guide for teachers - Years 7–8 June 2011 YEARS 78 Sets and Venn diagrams (Number and Algebra : Module 1) For teachers of Primary and Secondary Mathematics 510 Cover design, Layout design and Typesetting by Claire Ho The Improving Mathematics Education in Schools (TIMES) Project 2009‑2011 was funded by the Australian Government Department of Education, Employment and Workplace Relations. The views expressed here are those of the author and do not necessarily represent the views of the Australian Government Department of Education, Employment and Workplace Relations. © The University of Melbourne on behalf of the International Centre of Excellence for Education in Mathematics (ICE‑EM), the education division of the Australian Mathematical Sciences Institute (AMSI), 2010 (except where otherwise indicated). This work is licensed under the Creative Commons Attribution‑ NonCommercial‑NoDerivs 3.0 Unported License. 2011. http://creativecommons.org/licenses/by‑nc‑nd/3.0/ The Improving Mathematics Education in Schools (TIMES) Project NUMBER AND ALGEBRA Module 1 SETS AND VENN DIAGRAMS A guide for teachers - Years 7–8 June 2011 Peter Brown Michael Evans David Hunt Janine McIntosh Bill Pender Jacqui Ramagge YEARS 78 {1} A guide for teachers SETS AND VENN DIAGRAMS ASSUMED KNOWLEDGE • Addition and subtraction of whole numbers. • Familiarity with the English words ‘and’, ‘or’, ‘not’, ‘all’, ‘if…then’. MOTIVATION In all sorts of situations we classify objects into sets of similar objects and count them. This procedure is the most basic motivation for learning the whole numbers and learning how to add and subtract them.
    [Show full text]
  • On Contradictory Equivalence
    Daniil Teplitskiy, Russia, Saint Petersburg PSI Co. Ltd. 24JUN2011 To my teachers and good people Isaak Perchenok and Rustem Valeyev. To my friend Aleksey Kiselev whose intellectual paradigm underlies this work. On the scope of diagonal argument and on contradictory equivalence. Abstract : Restriction on the scope of diagonal argument will be set using two absolutely different proof techniques. One of the proof techniques will analyze contradictory equivalence (R ∈ R ↔ R ∉ R) in a rather unconventional way. Cantor’s paradox. Cantor’s paradox is based on two things: the first is Cantor’s theorem and the second one is the definition of U (universal set) as a set containing “everything” – all sets and all elements. Cantor’s theorem states that, for any set A, the set of all subsets of A (the power set of A) has a strictly greater cardinality than A itself. When we try to answer the question whose cardinality is larger, U or the power set of U, we can see from Cantor’s theorem that |power set of U|>|U| but according to the definition of U |U|≥|any other set| thus |U|≥|power set of U| . Of course, |power set of U|>|U| and |U|≥|power set of U| is a contradiction. The scope of Cantor’s theorem will be analyzed in both parts of the paper. Before starting the analysis I will give the plan of it: 1. At first Cantor’s theorem in its general form will be presented here. 2. Then an example of applying it to a more concrete thing – proof of uncountablity of power set of natural numbers will be presented.
    [Show full text]
  • Sections 2.1, 2.2 and 2.4 Chapter Summary
    SETS Sections 2.1, 2.2 and 2.4 Chapter Summary • Sets • The Language of Sets • Set Operations • Set Identities Introduction • Sets are one of the basic buildinggyp blocks for the types of objects considered in discrete mathematics. • Important for counting. • Programming languages have set operations. • Set theory is an important branch of mathematics. • Many different systems of axioms have been used to develop set theory. • Here we are not concerned with a formal set of axioms for set theory. Instead, we will use what is called naïve set theory. Sets • A set is an unordered collection of objects. • the students in this class • the chairs in this room • The o bjec ts in a se t are ca lle d the eltlements, or members of the set. A set is said to contain its elements. • The notation a ∈ A denotes that a is an element of the set A. • If a is not a member of A, write a ∉ A Describing a Set: Roster Method • S a,,,,b,c,d • Order not important S a,b,c,d b,c,a,d • Each distinct object is either a member or not; listing more than once does not change the set. S a,bdb,c,d a,bbdb,c,b,c,d • Elipses … may be used to describe a set without listing all of the members when the pattern is clear. S a,b,c,d, ……,z Roster Method • Set of all vowels in the Enggplish alphabet: V a,e,i,o,u • Set of all odd positive integers less than 10: O 1,3,5,7,9 • Set of all positive integers less than 100: S 1,2,3,……..,99 • Set of all integers less than 0: S …., ‐3,‐2,‐1 Some Important Sets N = natural numbers = ,,,0,1,2,3….
    [Show full text]